validation of goce models in africa
TRANSCRIPT
Validation of GOCE Models in Africa
Hussein A. Abd-Elmotaal
Civil Engineering Department, Faculty of EngineeringMinia University, Minia 61111, Egypt
Abstract
In modern geodesy, the geoid computation is carried out using the so-calledremove-restore technique. In this technique, the effect of the topography and itsisostatic compensation as well as the effect of a global reference field are removedfrom the gravimetric quantities being used in the geoid computation. The qualityof the reference geopotential model used in the framework of the remove-restoretechnique plays a great role in estimating the accuracy of the computed geoid. Inthe last decade, the GOCE gravity field dedicated satellite mission has taken placein order to enhance the gravity field recovery worldwide. Numerous GOCE geopo-tential models have already been released. Each geopotential model depends onthe strategy of the solution and the inclusion of terrestrial gravity data. The mainaim of this paper is to validate the recently released GOCE geopotential modelsin Africa. The results show that the EIGEN-6C4 and GO CONS GCF 2 DIR R5models give the smallest standard deviation and range of the reduced isostaticanomalies, respectively. However, the range and the standard deviation of thereduced isostatic anomalies are relatively too high, which shows that none of theGOCE released geopotential models ideally fits the African gravity field.
Keywords: Harmonic synthesis, geopotential models, Africa, gravity anomaly.
1. Introduction
The global geopotential models are widely used in practice in many geodetic appli-cations, such as gravity reduction, gravity smoothing, gravity interpolation and geoidcomputation. In the frame of the remove-restore geoid determination technique, theglobal geopotential models play an important role by introducing the effect of the longwavelength component of the gravity field. Hence, the proper choice of a global geopo-tential model in a geodetic application depends on how such a geopotential model fitsthe gravity field of the area of interest. The main aim of the current investigation is toexamine how suitable the recently released GOCE global geopotential models are to thegeodetic applications in Africa.
A number of recently available GOCE global geopotential models are consideredfor the current study. The local point free-air gravity anomaly data set for Africa isillustrated along with the available Digital Height Models necessary for the gravity re-duction computations. The used formulas to compute the synthesized gravity anomaliesfrom high-degree geopotential models are given. The used window remove-restore tech-nique (Abd-Elmotaal and Kuhtreiber, 2003) is outlined. The synthesized free-air gravityanomalies for Africa using the newly released GOCE global geopotential models are
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computed. The isostatic reduced gravity anomalies for Africa using the newly releasedGOCE global geopotential models are computed and widely discussed.
2. Used Geopotential Models
Table 1 illustrates the recently released GOCE global geopotential models used in thecurrent investigation. For the sake of completeness, the EGM2008 global geopotentialmodel is also used. Table 1 shows the models, their available maximum degree as wellas the data used to develop these models. The following abbreviations are adopted forthe used type of data in developing the geopotential models:
S = Satellite Tracking DataG = Gravity DataA = Altimetry Data
A wide range of geopotential models from satellite-only models with low maximum degreeto combined models with ultra high-degree are considered for the current study.
Table 1: Used global geopotential models
Model Year Degree Used Data Reference
EGM2008 2008 2160 S(Grace),G,A Pavlis et al., 2008, 2012
EIGEN-6C2 2012 1949 S(Goce,Grace,Lageos),G,A Forste et al., 2012
EIGEN-6C4 2014 2190 S(Goce,Grace,Lageos),G,A Forste et al., 2014
GO CONS GCF 2 TIM R3 2011 250 S(Goce) Pail et al., 2011
GO CONS GCF 2 TIM R5 2014 280 S(Goce) Brockmann et al., 2014
GO CONS GCF 2 DIR R5 2014 300 S(Goce,Grace,Lageos) Bruinsma et al., 2014
GOGRA02S 2013 230 S(Goce,Grace) Yi et al., 2013
GOGRA04S 2014 230 S(Goce,Grace) Yi et al., 2013
JYY GOCE04 2014 230 S(Goce) Yi et al., 2013
ITG-GOCE02 2013 240 S(Goce) Schall et al., 2014
3. The Data
3.1. Point Free-Air Gravity Anomalies
All currently available sea and land point free-air gravity anomalies for Africa are usedin the current investigation. Figure 1 shows their distribution. The distribution of thepoint free-air gravity anomaly stations on land is very poor. Many areas are empty. Thedistribution of the data points on sea is much better. More detailed information aboutthe free-air data for Africa, the technique used for the needed gross-error detection andthe merging technique can be found in (Abd-Elmotaal et al., 2015).
A total number of 1,186,032 free-air gravity anomaly values are available (cf. Fig. 1).The point free-air gravity anomalies range between −409.67 mgal and 452.80 mgal withan average of −4.37 mgal and a standard deviation of about 39.61 mgal.
In order to plot the free-air anomalies, they have been gridded on a 5′ × 5′ gridusing Kriging interpolation technique. Figure 2 shows the gridded 5′× 5′ free-air gravityanomalies for Africa used for the current investigation.
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Figure 1: Distribution of the point free-air land (green), shipborne (blue) andaltimetry derived (magenta) gravity anomalies in Africa.
3.2. Digital Height Models
For the terrain reduction computation, a set of fine and coarse Digital Height Models(DHM’s) is needed. The 30
′′ × 30′′SRTM30+ (Farr et al., 2007), which includes topog-
raphy and bathymetry, is employed as fine DHM, and the 3′ ×3
′SRTM is used as coarse
DHM. Figure 3 illustrates the 30′′ × 30
′′SRTM30+ fine DHM.
4. Basic Equations
The gravitational potential V can be expressed in spherical harmonic expansion as(Torge, 1989, p. 28; Dragomir et al., 1982, p. 53)
V (r, θ, λ) =GM
r
[1 +
∞∑n=2
(ar
)nn∑
m=0
(Cnm cosmλ+ Snm sinmλ
)Pnm(cos θ)
], (1)
where GM is the geocentric gravitational constant, r is the geocentric radius, θ is thepolar distance, λ is the geodetic longitude, a stands for the equatorial radius of themean earth’s ellipsoid, Pnm denotes the associated fully normalized Legendre functionsand Cnm and Snm are the fully normalized potential coefficients. The polar distance θcan simply be expressed in terms of the geocentric latitude ψ as:
θ = 90◦ − ψ , (2)
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Figure 2: Gridded 5′ × 5′ free-air gravity anomalies for Africa using thepoint gravity anomalies. Contour interval: 20 mgal.
-20 -10 0 10 20 30 40 50 60
-40
-30
-20
-10
0
10
20
30
40
-8000
-7000
-6000
-5000
-4000
-3000
-2000
-1000
0
1000
2000
3000
4000
5000
Figure 3: The fine 30′′ × 30
′′SRTM30+ DHM. Units in [m].
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where ψ is related to the geodetic latitude ϕ through the following expression (Torge,1980, p. 50):
tanψ = (1− f)2 tanϕ , (3)
where f is the flattening of the earth’s ellipsoid.The disturbing potential T is defined by
T (r, θ, λ) = V (r, θ, λ)− U(r, θ) (4)
where U is the normal gravitational potential of the mean earth’s ellipsoid, given by(Torge, 1989, p. 37)
U(r, θ) =GM
r
[1 +
∞∑n=2
(ar
)n
Cun0Pn0(cos θ)
]. (5)
Here Cun0 denotes the fully normalized harmonic coefficients implied by the reference
equipotential ellipsoid. Because of the rotational symmetry of the mean earth’s ellip-soid, there will be only zonal terms. And because of the symmetry with respect to theequatorial plane, there will be only even zonal harmonics Cu
2n,0 (Heiskanen and Moritz,1967, p. 72). The even degree zonal harmonic coefficients Cu
2n,0 converge quickly towardzero, so that (5) may safely be truncated after n = 6.
Thus inserting (1) and (5) into (4), the disturbing potential T can be expressed as(Torge, 1989, p. 43)
T (r, θ, λ) =GM
r
∞∑n=2
(ar
)nn∑
m=0
(C
∗
nm cosmλ+ Snm sinmλ)Pnm(cos θ) , (6)
where C∗nm is the difference between the actual coefficients Cnm and those implied by
the reference equipotential ellipsoid Cunm.
The gravity anomaly ∆g can be expressed, using the spherical approximation, by(Moritz, 1980)
∆g(r, θ, λ) =GM
r2
∞∑n=2
(n− 1)(ar
)nn∑
m=0
(C
∗
nm cosmλ+ Snm sinmλ)Pnm(cos θ) . (7)
5. The Window Technique
The traditional way of removing the effect of the topographic-isostatic masses faces atheoretical problem. A part of the influence of the topographic-isostatic masses is re-moved twice as it is already included in the global reference field. This leads to somedouble consideration of that part of the topographic-isostatic masses. Figure 4 sketchesthe traditional gravity reduction for the effect of the topographic-isostatic masses. Theshort-wavelength part, depending on the topographic-isostatic masses, is computed fora point P for the masses inside the circle denoted by TC. Removing the effect of thelong-wavelength part by a global earth’s gravitational potential field normally impliesremoving the influence of the global topographic-isostatic masses, shown as the big rect-angle in Fig. 4 denoted by EGM (here EGM stands for Global Geopotential Model).
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a
P
EGM
TC
.
Figure 4: The traditional remove-restore technique.
The double consideration of the topographic-isostatic masses inside the circle (doublehatched) is thus seen.
A possible way to overcome this difficulty is to adapt the used reference field tothe effect of the topographic-isostatic masses for a fixed data area (Abd-Elmotaal andKuhtreiber, 2003). Figure 5 shows the advantage of the window remove-restore tech-nique. Consider a measurement at point P ; the short-wavelength part, depending onthe topographic-isostatic masses, is now computed by using the masses of the whole dataarea (the small rectangle in Fig. 5). The adapted reference field is created by subtractingthe effect of the topographic-isostatic masses of the data window, in terms of potentialcoefficients, from the reference field coefficients. Thus, removing the long-wavelengthpart by using this adapted reference field does not lead to a double consideration of apart of the topographic-isostatic masses (as there is no double hatched area in Fig. 5).
a
adapted EGM
P
data area
TC.
Figure 5: The window remove-restore technique.
The remove step of the window remove-restore technique can then mathematicallybe written as
∆giso win = ∆gF −∆gTI win −∆gGM Adapt , (8)
where ∆giso win stands for the isostatic anomalies computed using the window technique,∆gGM Adapt is the contribution of the adapted reference field and ∆gTI win is the effectof topography and its compensation for the fixed data window on the gravity anomaliesemploying the fine and coarse DHM’s described in sec. 3.2.. Equation (8) can be writtenmore explicitly as
∆giso win = ∆gF −∆gTI win − (∆gGM −∆gwincof )
= ∆gF −∆gTI win −∆gGM +∆gwincof , (9)
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where ∆gGM is the contribution of the original global reference field and ∆gwincof is thecontribution of the harmonic coefficients of the topographic-isostatic masses of the datawindow (for our case it is the African data window −40◦ ≤ ϕ ≤ 42◦, −20◦ ≤ λ ≤ 60◦)computed to the maximum degree of the used geopotential model.
The computation of the potential harmonic coefficients of the topographic-isostaticmasses within the data window is carried out using (Abd-Elmotaal and Kuhtreiber,1999, Eq. (11)). The contribution ∆gwincof is computed from the potential harmoniccoefficients of the topographic-isostatic masses within the data window (for our case itis the African data window). For more details on the window remove-restore technique,the reader is kindly invited to refer to (Abd-Elmotaal and Kuhtreiber, 1999, 2003).
Let us define a couple of terms. The first term is called “the long-medium wavelengthfree-air anomalies” ∆gF long-med, which is defined as
∆gF long-med = ∆gF −∆gTI win +∆gwincof . (10)
The second term is called “the reduced free-air anomalies” ∆gF red, which is defined as
∆gF red = ∆gF −∆gGM . (11)
6. The Results
Table 2 shows the statistics of the point free-air gravity anomalies as well as of the free-air anomalies produced by the used global geopotential models. The program GRVHRM(Abd-Elmotaal, 1998) has been used to compute the free-air gravity anomalies at thegravity data points produced by the used geopotential models ∆gGM .
Table 2: Statistics of the free-air gravity anomalies produced by the usedglobal geopotential models (1,186,032 stations). Units in [mgal]
gravity anomalies Nmax min. max. mean st. dev.
Point free-air (∆gF ) — −409.67 452.80 −4.37 39.61
∆gF long-med — −237.16 322.42 −3.28 36.30
EGM2008 2160 −234.72 578.21 −2.51 38.14
EIGEN-6C2 1949 −237.06 573.60 −2.52 37.97
EIGEN-6C4 2190 −233.70 579.57 −2.50 37.96
GO CONS GCF 2 TIM R3 250 −191.72 176.16 −0.86 28.26
GO CONS GCF 2 TIM R5 280 −185.04 190.75 −0.96 29.11
GO CONS GCF 2 DIR R5 300 −182.12 190.64 −1.02 29.35
GOGRA02S 230 −197.34 176.59 −0.93 28.27
GOGRA04S 230 −196.16 176.96 −0.81 28.24
JYY GOCE04S 230 −196.06 176.90 −0.81 28.24
ITG-GOCE02 240 −190.47 175.88 −0.69 27.97
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According to Eq. (9), in order to get as much smaller reduced anomalies as possible,the term “the long-medium wavelength free-air anomalies” ∆gF long-med should compen-sate with the term ∆gGM . Accordingly, Table 2 may give the sign that none of the ex-amined geopotential models would satisfy this criterion. The range of the long-mediumwavelength free-air anomalies ∆gF long-med is only 65% of that of the free-air anomaliesand the standard deviation is about 92% of that of the free-air anomalies (which signal-izes that the short wavelength are mostly removed from the free-air anomalies). Table 2shows that the ultra high-degree models have generally greater range than that of thelong-medium wavelength free-air anomalies ∆gF long-med with slightly higher standarddeviation. In contradictory, the satellite-only models (which have low upper maximumdegree) have generally much smaller range than that of the long-medium wavelengthfree-air anomalies ∆gF long-med with significantly smaller standard deviation.
Table 3 shows the statistics of “the reduced free-air anomalies” ∆gF red for all usedglobal geopotential models. No topographic-isostatic reduction has been made at thisstage. Table 3 shows that the range of the reduced free-air gravity anomalies ∆gF red forall models is at the same order of magnitude as that of the point free-air gravity anoma-lies. It shows also that the standard deviation of the reduced free-air anomalies ∆gF red
for the ultra high-degree models is dramatically smaller than that of the satellite-onlymodels. This is simply because the satellite-only models could not remove the mediumwavelength from the free-air anomalies having a relatively low upper maximum degree.This may also come from the possible inclusion of some of the gravity measurements overAfrica in the creation of the ultra-high geopotential models. Table 3 also shows that themean value of the reduced free-air anomalies ∆gF red for the ultra high-degree models issmaller than that of the satellite-only models.
Table 3: Statistics of the reduced free-air gravity anomalies ∆gF red computed by Eq. (11)for the used global geopotential models (1,186,032 stations). Units in [mgal]
used geopotential model Nmax min. max. mean st. dev.
EGM2008 2160 −457.03 271.53 −1.86 13.24
EIGEN-6C2 1949 −473.32 272.07 −1.85 13.47
EIGEN-6C4 2190 −456.51 270.80 −1.88 13.35
GO CONS GCF 2 TIM R3 250 −525.32 313.08 −3.51 29.10
GO CONS GCF 2 TIM R5 280 −527.16 294.74 −3.41 27.94
GO CONS GCF 2 DIR R5 300 −525.37 295.34 −3.35 27.68
GOGRA02S 230 −515.18 323.29 −3.44 29.22
GOGRA04S 230 −510.40 320.69 −3.56 29.09
JYY GOCE04S 230 −510.44 320.57 −3.56 29.09
ITG-GOCE02 240 −518.39 316.61 −3.68 29.41
Figure 6 shows the free-air reduced gravity anomalies ∆gF red for the EIGEN-6C4geopotential model, being one of the best models to generate the reduced free-air anoma-lies for Africa. It should be noted that EGM2008 model gives nearly the same statisticsof the reduced free-air gravity anomalies. Figure 6 shows that most of the area has
156
free-air reduced gravity anomalies less than 20 mgal in magnitude (the white pattern).More than 91.4% of the points have reduced free-air anomalies less than 20 mgal inmagnitude. The higher values are located at the high mountainous area of Morocco,which is likely due to vertical datum inconsistency. Deeper investigation for this area isessentially needed (cf. Abd-Elmotaal et al., 2015).
Figure 6: The EIGEN-6C4 reduced free-air anomalies ∆gF red for Africa(computed by Eq. (11)).
Table 4 shows the statistics of the isostatic anomalies computed using the windowtechnique ∆giso win for the used geopotential models. The Airy-Heiskanen isostatic modelhas been used with the following parameters:
T◦ = 30 km ,
ρ◦ = 2.67 g/cm3 , (12)
∆ρ = 0.4 g/cm3 .
Table 4 shows that the standard deviation and the mean values of the Airy window iso-static gravity anomalies for the ultra high-degree models are slightly smaller than thoseof the satellite-only models. The range of the Airy window isostatic gravity anoma-lies for the ultra high-degree models is, however, significantly larger than that of thesatellite-only models. Generally, the range and the standard deviation of the Airy win-dow isostatic anomalies are relatively too high for such kind of gravity anomalies, whichmay signalize that none of these models is ideal for geodetic applications in Africa.
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More interesting is the significant drop of the standard deviation of the the Airy win-dow isostatic gravity anomalies for the satellite-only models. This already indicates animprovement, but perhaps not to the desired extent.
Table 4: Statistics of the isostatic anomalies computed using the window technique∆giso win using Eq. (9) for the used global geopotential models
(1,186,032 stations). Units in [mgal]
used geopotential model Nmax min. max. mean st. dev.
EGM2008 2160 −423.93 270.86 −0.77 20.09
EIGEN-6C2 1949 −409.02 271.40 −0.77 20.03
EIGEN-6C4 2190 −423.59 270.13 −0.79 20.00
GO CONS GCF 2 TIM R3 250 −202.96 282.96 −2.42 22.86
GO CONS GCF 2 TIM R5 280 −207.79 281.82 −2.33 21.55
GO CONS GCF 2 DIR R5 300 −205.08 276.98 −2.26 21.22
GOGRA02S 230 −212.28 281.00 −2.35 23.10
GOGRA04S 230 −210.06 284.15 −2.47 23.02
JYY GOCE04S 230 −209.96 284.10 −2.47 23.02
ITG-GOCE02 240 −207.72 285.43 −2.59 23.33
In order to distinguish between the case on land and on sea regions, two tables areestablished. Tables 5 and 6 show the statistics of the isostatic anomalies computedusing the window technique ∆giso win for the used geopotential models on land and sea,respectively. Generally, the Airy window isostatic anomalies on land have slightly higherstandard deviation and range than those on sea for all models.
Table 5: Statistics of the isostatic anomalies on land computed using the windowtechnique ∆giso win using Eq. (9) for the used global geopotential models
(94,838 stations). Units in [mgal]
used geopotential model Nmax min. max. mean st. dev.
EGM2008 2160 −423.93 270.86 −0.39 23.65
EIGEN-6C2 1949 −409.02 271.40 −0.34 23.59
EIGEN-6C4 2190 −423.59 270.13 −0.19 23.70
GO CONS GCF 2 TIM R3 250 −202.96 282.96 −0.27 25.38
GO CONS GCF 2 TIM R5 280 −207.79 281.82 −0.52 23.96
GO CONS GCF 2 DIR R5 300 −205.08 276.98 −0.53 23.62
GOGRA02S 230 −212.28 281.00 −0.14 25.81
GOGRA04S 230 −210.06 284.15 −0.14 25.68
JYY GOCE04S 230 −209.96 284.10 −0.14 25.68
ITG-GOCE02 240 −207.72 285.43 −0.16 25.77
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Table 6: Statistics of the isostatic anomalies on sea computed using the windowtechnique ∆giso win using Eq. (9) for the used global geopotential models
(1,091,194 stations). Units in [mgal]
used geopotential model Nmax min. max. mean st. dev.
EGM2008 2160 −187.90 142.50 −0.81 19.75
EIGEN-6C2 1949 −186.05 145.02 −0.80 19.69
EIGEN-6C4 2190 −187.65 143.68 −0.84 19.65
GO CONS GCF 2 TIM R3 250 −152.32 159.09 −2.61 22.62
GO CONS GCF 2 TIM R5 280 −140.64 159.21 −2.48 21.32
GO CONS GCF 2 DIR R5 300 −140.02 151.68 −2.41 21.00
GOGRA02S 230 −149.29 167.32 −2.55 22.84
GOGRA04S 230 −149.01 168.93 −2.67 22.76
JYY GOCE04S 230 −149.08 168.99 −2.68 22.76
ITG-GOCE02 240 −153.41 170.45 −2.80 23.09
Figure 7 shows the Airy window isostatic gravity anomalies ∆giso win for the EIGEN-6C4 geopotential model, being the best model giving the smallest standard deviationof the Airy window isostatic anomalies. Figure 7 shows that most of the area has Airywindow isostatic gravity anomalies less than 20 mgal in magnitude (the white pattern).More than 78.5% of the points have Airy window isostatic anomalies less than 20 mgalin magnitude.
Figure 8 shows the Airy window isostatic gravity anomalies ∆giso win for the satellite-only GO CONS GCF 2 DIR R5 geopotential model, which have a significantly betterrange than that of EIGEN-6C4 model (more than 210 mgal less) and a slightly largerstandard deviation (1.2 mgal higher). The advantage of the GO CONS GCF 2 DIR R5geopotential model is that its upper maximum degree is only 300, which saves a tremen-dous amount of CPU time for computing the synthesized gravity anomalies ∆gGM . Fig-ure 8 shows that most of the area has Airy window isostatic gravity anomalies less than20 mgal in magnitude (the white pattern). More than 75.2% of the points have Airywindow isostatic anomalies less than 20 mgal in magnitude.
7. Conclusion
Different recent GOCE geopotential models are tested to produce reduced isostaticgravity anomalies for Africa. The reduction of the gravity anomalies follows the win-dow remove-restore technique employing the Airy floating hypothesis. A wide range ofgeopotential models are tested from satellite-only models, having low upper degree, tocombined models having ultra high upper degree.
The results show that the GOCE-GRACE-LAGEOS combined geopotential modelEIGEN-6C4 gives the smallest standard deviation of the Airy window isostatic anomaliesfor Africa. The GOCE satellite-only model GO CONS GCF 2 DIR R5 gives the smallestrange of the Airy window isostatic anomalies for Africa, with only 1 mgal higher in the
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Figure 7: The EIGEN-6C4 Airy window isostatic anomalies ∆giso win for Africa(computed by Eq. (9)).
standard deviation compared to that of the EIGEN-6C4 model. Both the range andstandard deviation of the Airy window isostatic anomalies generated by all tested modelsare relatively high.
It has been shown that most of the points on land in Africa have Airy windowisostatic anomalies below 20 mgal in magnitude. There are still some spots, like in thehigh mountainous area of Morocco, that they still have large values of the Airy windowisostatic anomalies. This needs, however, a deeper investigation. We believe that usingsome of the African land gravity measurement in the scaling process of the satellite-onlymodels would definitely increase their fit to the African gravity field.
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